Categories of Commutative Algebras

Hardcover | April 30, 1999

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This book studies the universal constructions and properties in categories of commutative algebras, bringing out the specific properties that make commutative algebra and algebraic geometry work. Two new universal constructions are presented and used here for the first time. The author showsthat the concepts and constructions arising in commutative algebra and algebraic geometry are not bound so tightly to the absolute universe of rings, but possess a universality that is independent of them and can be interpreted in various categories of discourse. This brings new flexibility toclassical commutative algebra and affords the possibility of extending the domain of validity and the application of the vast number of results obtained in classical commutative algebra.

About The Author

Yves Diers is in the Department de Mathematiques, I.S.T.V Universite de Valenciennes.

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'reduced schemes correspond to schemes on the Zariski category RedCRing of reduced commutative rings, etc., thus making the notion of a scheme even more natural in the present context. The book formalizes this point of view in a very elegant way, providing a wide variety of well-chosenexamples to make it attractive to a broad, mixed audience. The last chapter provides methods to construct new Zariski categories from known ones, proving, once again, the universality and wide applicability of the techniques covered in this book.'A. Verschoren, Mathematics Abstracts, 772/93